On the Functional Independence of Zeta-Functions of Certain Cusp Forms

The zeta-function zeta(s, F), s = sigma + it of a cusp form F of weight kappa in the half-plane sigma > (kappa + 1)/2 is defined by the Dirichlet series whose coefficients are the coefficients of the Fourier series of the form F. The compositions V(zeta(s,F)) with an operator V on the space of analytic functions are considered, and the functional independence of these compositions for certain classes of operators V is proved.